Monomial ideals with linear quotients and componentwise (support-)linearity

TL;DR

This paper investigates monomial ideals with linear quotients, demonstrating their componentwise support-linearity, introducing variable decomposable ideals, and exploring their relationships with other special classes of ideals.

Contribution

It provides new proofs of support-linearity for ideals with linear quotients and introduces variable decomposable ideals, linking them to known classes like vertex decomposable complexes.

Findings

Monomial ideals with linear quotients are componentwise support-linear.

Variable decomposable ideals generalize vertex decomposable complexes.

Relationships between variable decomposable, weakly polymatroidal, and weakly stable ideals are established.

Abstract

When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable decomposable monomial ideals. In squarefree case, these ideals correspond to the vertex decomposable simplicial complexes. We study the relationships of the variable decomposable ideals with weakly polymatroidal ideals, weakly stable ideals and ideals with linear quotients. We also investigate the componentwise properties of all these ideals.